4 months ago, thinking on balance puzzles like these, I started wondering about problems of imbalance. “What if the scales tip?” I thought, “what information do we get, and how can I use it to make a good puzzle?” I wrote fifteen imbalance puzzles of my own, but I took things further and offered prizes to my two favorite imbalance puzzle writers. In the math section above, I put together a page including my puzzles and the contest submissions. You should take a look, because there are LOTS of good submissions.

BUT four months is plenty, and it’s time to announce winners. Congratulations to Nathan Chow and David Price who I’ve selected as my favorite puzzlists!!! Based, as I said, on “incredibly subjective criteria,” I chose these two for the way that they extended the state of the art. Their problems really stretched my ideas into wonderful new territory. THANKS BE TO THEM!

As promised, Nathan and David each win a print of their choosing from my Stars of the Mind’s Sky series, up to 13″x 13″. All of the rest of us get to solve these wonderful puzzles.

Honorable mention goes to Felix, a fifth grader I got to work with last year. Felix is a wonderful young mathematician, and he came up with a really nifty imbalance problem. I don’t want to spoil it for you so solve now, and I’ll continue below.

Felix (5th grade)

SPOILER: Felix started with the idea that his puzzle would include negative weights, which really tickled him. I think he reveals the information really nicely in the puzzle. Thanks, Felix, for the wonderful puzzle!

I’m really happy to see how many people have been excited by these imbalance problems. The entries continue to roll in for my imbalance problem writing contest. You can check them all out here, and I’ll continue to update that page. I’m so happy to see other people’s ideas stretch my thinking and help me see what’s possible. This is just really fun. I have some early favorites, but I won’t give anything away just yet. Hopefully I’ll get more submissions. maybe YOU will make one.

I’m noticing all sorts of techniques for solving them. In fact, I learned something from a recent submission and it inspired three new puzzles of my own! I hope you enjoy. I’d love your feedback.

I’m really digging these imbalance problems I came up with. Friday was our last day before spring break, and I worked on them with every one of my classes. They liked the puzzles, but the examples I had were a bit too easy, so our real goal was creating good imbalance problems of our own, which is way harder than solving them. My 5th graders had the best ideas for tweaking the puzzles and making new ones. It was cool for them to realize you couldn’t just draw a picture, because sometimes they were impossible, and other times they didn’t give enough information. I’ll hopefully post some of their puzzles after break.

I’ve been working the last two days on my own puzzles, and I’ve never had a better time working with inequalities in my life. They have a way of sneaky way of revealing information that I’m really liking. 2x<y+z tells you some interesting stuff, for example. Further down I’ll talk about how I write them, what makes for a good puzzles, and the puzzle-writing contest I’m having, but right now, why don’t you try some out? I’m especially proud of 6, 9, 10, 11, and 12. ENJOY!

In each case, order the three shapes by weight

Fun, right? As I’ve said, I’m offering a prize for great imbalance puzzle-writing. My favorite two puzzlists will receive a print of their choosing from the Stars of the Mind’s Sky series, up to 12″x12″. Just post your problem(s) in the comments or email lostinrecursion@gmail.com.

So what makes a good puzzle? It’s a matter of taste (like a lot of mathematics actually), but I tend to like my puzzles pretty simple. A big, messy, just plain hard puzzle is just a hassle, but something compact and tricky to untangle, now that’s what I like. I love when a puzzle requires me to think in a new way or exploit some clever little detail I hadn’t considered. Nathan Chow sent me a really clever puzzle design that implements “entangled imbalances,” which I completely love. Doesn’t it inspire you to write your own? I could have cropped it so it didn’t look like an iPhone app, but I love art that reveals the creative process.

In case you’re wanting tips for writing these puzzles, I’ll tell you about my process. I usually start with a single idea or part of the picture. The key is thinking about what information it gives the solver, and what other information they’ll need to finish the problem. After that it’s just a matter of cleverly revealing that information and piecing them together. Maybe solving the problems above in order will give you a sense of the new ideas I had and was able to wrinkle in.

I hope you’re loving these as much as I am, and I’m dying to see your creations. Mostly because I want some to solve!!!

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Paul is a mathematician with a Masters from CUNY - Hunter College. He teaches math at John Burroughs School, in St. Louis. He's also a mathematical artist whose work has exhibited internationally. Most proudly, though, Paul is a father and husband to Nora and Teresa Feathers.